Scaling Relations for Dark Matter Halos Hosting Ultra-Faint Dwarf Galaxies

We consider the extraction of parameters of dark matter halos hosting ultra-faint dwarf galaxies, in the case where there are only ${\cal O}(10)$ identified member stars with measured line-of-sight velocities. This scenario is likely to be increasingly common, as upcoming newly discovered dwarf galaxies in the Milky Way, by e.g. the Rubin Observatory, will likely (at least initially) have only a few identified members. Assuming an NFW dark matter profile, equilibrium modeling likely can only robustly extract one halo parameter ($\rho_s r_s$), but the scale radius itself will typically be unconstrained. In these cases, the results obtainable from Jeans modeling can be well replicated by a simple scaling relation motivated by the half-light mass estimator. As a application, we examine the recently discovered stellar system Ursa Major III, which has been optimistically assessed to have the largest $J$-factor of any known object. We suggest that, because of the presence of outlier stars, the $J$-factor obtained from modeling of Ursa Major III is likely inflated, as it is inconsistent with the half-light mass estimator, while removal of the outliers will leave the $J$-factor unconstrained from below.